Post on 14-Jan-2020
Fujita LaboratoryTokyo Institute of Technology
Tokyo Institute of Technology
Visual Feedback for Cooperative Motion Control
ICCAS-SICE 2009 Tutorial Mini-Course9:00 – 11:00, August 21st, 2009
Masayuki FujitaDepartment of Mechanical and Control Engineering
Tokyo Institute of Technology, JapanFujita LaboratoryTokyo Institute of Technology
Tokyo Institute of Technology
Outline
・ Relative Rigid Body Motion
・ Prologue – Cooperative Control
・ Pose Control with VMO
・ Visual Motion Observer (VMO)
・ Robot Control with VMO
・ Epilogue – Cooperative Control with VMO
2
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Agent i’s System Dynamics : State: Output: Input
Agent i’s Storage Function
Passivity
Fig.1: Block Diagram of Agenti’s System Dynamics
Passive
Storage FunctionSystem Dynamics : State: Output: Input
Passivity
(1)
(2) (3)
(4)
(7)
(5) (6)
Definition of Passivity
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(12)
4
Closed-loop Systems
Networked Passive Systems
Passive
Fig. 2: Networked Passive Systems
Networked Passive Systems (Three Agents)
Fig. 3: Simulation Result
SynchronizationOutput
Control Input: Output Error with Neighbors
Fig. 4: Networked Passive Systems in SE(3)
SE(3)
(8) (9) (10)
(14)
(11) (13)
Lyapunov Function Candidate
Sum of Individual Storage Functions(17)
(18)
(15) (16)
4
ICCAS-SICE 2009 Tutorial Mini-Course August 21, 2009, Fukuoka International Congress Center, Fukuoka, JAPAN
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Position (Translation):
Orientation (Rotation):
Homogeneous Transformation
5
Fig. 5: Homogeneous Transformation
Homogeneous Transformation
“∧” (wedge) : “∨” (vee) :
: Rotation Axis: Rotation Angle
Exponential Coordinate for Rotation
Fig. 6: Exponential Coordinatefor Rotation
(19) (20)
(23)
(21) (22)
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Rigid Body Motion in SE(3)
: Position
: Orientation
Fig. 9: Block Diagram of Rigid Body Motion
Rigid BodyMotion
(24)
Pose (Position and Orientation)
(27)Rigid Body Motion
:Velocity of relative to as viewed in the current wedge
vee(25)
(26)
Body Velocity
Fig. 7: Body Velocity
(26)
Fig. 8: Coordinate of Rigid Body in SE(3)
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Passivity of Rigid Body Motion
7
The rigid body motion (27) satisfies
where is a positive scalar.
Lemma 1
Passive
(31)
Rigid BodyMotion to
Vector Representation of Pose(Position)
(Orientation)(28)
: Coordinate Transformationof Vector from to
( to )
to Skew-symmetric Component
(29) (30)
Fig. 10: Block Diagram of Passivity of Rigid Body Motion
Fig. 8: Coordinate of Rigid Body in SE(3)
(27)Rigid Body Motion
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(33)Attitude Synchronization
Consider the n rigid bodies represented by (27). Then a group of rigid bodies is said to achieve attitude synchronization, when all rigid bodies converge to the same orientation between the rigid bodieswhile moving in the same direction.
Attitude Synchronization
Error Function of Rotation Matrix Property :・
・
・
Attitude Synchronization
Relative Orientation
(32)
(34)Fig. 11: Attitude Synchronization
j-th rigid body’s orientation as viewed in
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Fig. 12: Network Topology
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Control Input for Attitude Synchronization
Control Input for Attitude Synchronization
(36)
Linear Velocity Input
Angular Velocity Input
Relative Orientation
(35)
Fig. 13: Block Diagram of Attitude Synchronization
i-th Attitude Synchronization Controller
i-th RigidBody Motion
j-th RigidBody Motion
Neighborhood : A set of agents whose information is available to agent i
Fig. 11: Attitude Synchronization
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Attitude Synchronization
Theorem 1Consider the n rigid bodies represented by (27). Then, under the assumptions A1 and A2, the velocity input (35), (36) achieves attitude synchronization in the sense of (33).
There is a directly path connecting any two distinct nodes.
All agents are able to get any agent’s information direct or indirectly.
Strongly Connected
Assumptions(A1)(A2) The graph is strongly connected and fixed
are positive definite
: positive definite
Fig. 12: Network Topology
Fig. 11: Attitude Synchronization
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Outline
・ Relative Rigid Body Motion
・ Prologue – Cooperative Control
・ Pose Control with VMO
・ Visual Motion Observer (VMO)
・ Robot Control with VMO
・ Epilogue – Cooperative Control with VMO
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Relative Pose and Body VelocityPose of Vision Camera
Pose of Object(37)
(38)Pose of Object relative to
(39)
(40)
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Body Velocity of Vision Camera
Body Velocity of Object(41)
(43)
Body Velocity of Object relative to
(45)
(42)
(46)
(44)
Vision Camera
Vision Camera
: Position
: Orientation
Object Frame World Frame
Vision Camera Frame
Fig. 14: Relative Pose and Body Velocity of Object relative to Vision Camera in SE(3)
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Relative Rigid Body Motion
Differentiating (40) w.r.t. time(40)Pose of Object relative to Vision Camera:
(47)
=(47)
(48)
=(42) =(44)
Fig. 14: Relative Pose and Body Velocity of Object relative to Vision Camera in SE(3)
Object Frame World Frame
Vision Camera Frame
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vee(49)
Adjoint Transformation
(50)Relative Rigid Body Motion
Body Velocity ofVision Camera
(52)Body Velocityof Object
: Coordinate Transformationfrom to
Relative Rigid Body Motion
Fig. 15: Block Diagram of Body Velocity of Object relative to Vision Camera
+−
(48)
Body Velocity of Object relative to
(53)
(51)
Vision Camera
Fig. 14: Relative Pose and Body Velocity of Object relative to Vision Camera in SE(3)
Object Frame World Frame
Vision Camera Frame
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Vector Representation of Pose
Fig. 16: Block Diagram of Vector Representation of PoseRelative Rigid Body Motion
(40)
Vector Representation of Pose(Position)
(Orientation)(55) to
( to )
to
Vector Representation of Rotation Matrix
MatrixVector
(54)
Skew-symmetric Component
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Passivity of Relative Rigid Body Motion
Lemma 2
(56)
where is a positive scalar.
If the object is static , then the relative rigid body motion (52) satisfies
Rotation Matrix
Property :・
・
・
Error Function of
(58)Storage Function
(57)
Relative Rigid Body Motion
Skew-symmetric Matrix
Differentiating (58) w.r.t. time yields(Proof)
(59)
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(Q.E.D.)
Integrating (60) from 0 to T, we obtain
where is a positive scalar that only depends on the initial states of .
(61)
PassiveRelative RigidBody Motion to
Fig. 17: Block Diagram of Passivity of Relative Rigid Body Motion
Passivity of Relative Rigid Body Motion
(32)Property of Skew-symmetric
(60)
Remark 1
motion (52) satisfies where is aIf the vision camera is static , then the relative rigid body
Passivepositive scalar.Fujita LaboratoryTokyo Institute of Technology
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Outline
・ Relative Rigid Body Motion
・ Prologue – Cooperative Control
・ Pose Control with VMO
・ Visual Motion Observer (VMO)
・ Robot Control with VMO
・ Epilogue – Cooperative Control with VMO
18
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Feature Points
Fig. 19:Pinhole Camera Model
: not measurable
World FrameObject Frame World FrameUnknown
Vision CameraFrame
not measurable
Unknown
Relative RigidBody Motion
Fig. 18: Block Diagram of Relative Rigid Body Motion with Vision Camera
Object’s i-th Feature Point
PointFeature
(63)
i-th Feature Point
PointsFeature
(62)
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(64)
Image Information (m Points)
(65)
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Perspective Projection
Perspective Projection (Pinhole Camera)
Fig. 19:Pinhole Camera ModelUnknown
not measurableVision Camera
measurable
PointsFeature
ProjectionPerspective
Unknown
Relative RigidBody Motion
Fig. 18: Block Diagram of Relative Rigid Body Motion with Vision Camera
World FrameObject Frame World Frame
Vision CameraImage Plane
: Focal Length
Frame
PointFeature
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Relative Rigid Body Motion (RRBM)
Actual Posenot measurable
measurableImage Information
Luenberger Observer
(64) (65)
(52)
RRBM Camera
not measurable measurable
Vision
Unknown
Fig. 20: Block Diagram of Estimation Error Vector
(66):Input for Estimation Error
Relative Rigid Body Motion (RRBM) Model
Estimated Image Information
Estimated Pose
(67) (68)
to ControlInput
estimated
ModelCamera
estimated
RRBM Vision
Model21
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Estimation Error
Estimation Error Vector
(69)(Error between Estimated State and Actual One)
Estimation Error
(Position)(Orientation)(70)
Image Information Error(71)
(Actual) ImageInformation
Estimated Image Information
Fig. 20: Block Diagram of Estimation Error Vector
estimated
ModelCamera
RRBM Camera
not measurable measurable
estimated
RRBM
Vision
Vision
Model
Unknown
Relation between the actual image information and the estimated one
: i-th Image Jacobian(72)
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Estimation Error System
Fig. 20: Block Diagram of Estimation Error Vector
estimated
ModelCamera
RRBM Camera
not measurable measurable
estimated
RRBM
Vision
Vision
Model
Estimation Error(69)
Estimation Error Vector
(70)Image
Jacobian
(73)Image
Information
Estimation error can be calculatedusing image information !!
PoseInformation
(75)
Estimation Error System(74)
System
estimated
ModelCamera
RRBM Camera
not measurable measurable
estimated
RRBM
ErrorEstimation
Vision
Vision
Model23
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Passivity of Estimation Error System
Lemma 3
(76)where and is a positive scalar.
If the object is static , then the estimation error system (75) satisfies
Passive
Fig. 21: Block Diagram of Passivity of Estimation Error System
System
estimated
ModelCamera
RRBM Camera
not measurable measurable
estimated
RRBM
ErrorEstimation
Vision
Vision
Model
Passive
Skew-symmetric Matrix
(Sketch of Proof)
(78)
Storage Function
(77)
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Fig. 22: Block Diagram of Visual Motion Observer
Control Law for Visual Motion Observer : Passivity Approach
Visual Motion Observer
(79)Theorem 2If , then the equilibrium point for the closed-loop system (75) and (79) is asymptotic stable.
Lyapunov Function Candidate(77)
(80)System
estimated
ModelCamera
RRBM Camera
not measurable measurable
estimated
RRBM
ErrorEstimation
Vision
Vision
Model
Fig. 23: Block Diagram of Visual Motion Observer
MotionVisual
Observer
Visual Motion Observer
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Outline
・ Relative Rigid Body Motion
・ Prologue – Cooperative Control
・ Pose Control with VMO
・ Visual Motion Observer (VMO)
・ Robot Control with VMO
・ Epilogue – Cooperative Control with VMO
26
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Fig. 25: Block Diagram of Relative Pose Computation
Pose ControlBring the relative poseto the desired pose .
× : not measurable only from f
Unknown Fig. 24:Pose Control ofEye-in-HandSystem
ConstantDesiredPose
Pose Control of Eye-in-Hand System
MotionVisual
Observer
×(81) (69)
to
Pose Computation
( to )to
○(81)
If Attitude Estimation Error
Vector Matrix(82)
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Pose Control Error System
(85)
Pose Control Error System
Pose Control Error
Pose Control Error Vector
(83)
(84)
Unknown
Passive
(This is dual to the estimation error system.)
Desired Pose :Constant
Fig. 26: Block Diagram of Pose Control Error System
Pose Computation
toto
Pose ControlError System
to
Fig. 24:Pose Control ofEye-in-HandSystem
MotionVisual
Observer
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Fig. 27: Block Diagram of Error System for Pose Control with Visual Motion Observer
Error System for Pose Control with Visual Motion Observer
Pose Control Error System
estimated
measurable
Estimation Error System
(86)
State Input Disturbance
Error System for Pose Control with Visual Motion Observer
Pose Control & Estimation
Pose Control Error System
Estimation Error System+
totoRRBM& Camera
Model
Error Systems
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Fig. 27: Block Diagram of Error System for Pose Control with Visual Motion Observer
Pose Control & Estimation Error Systems
Passivity of Error System for Pose Control with Visual Motion Observer
Error SystemPose Control
Error SystemEstimation
Lemma 4If the object is static , then the error system forpose control with the visual motion observer (86) satisfies
(87)
where , and is a positive scalar.
Passive
Passive
Storage Function(88)
Skew-symmetric Matrix
(Sketch of Proof)(89)
(90)
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Fig. 28: Block Diagram of Pose Control with Visual Motion Observer
Pose Control Law
Pose Control Law with Visual Motion Observer : Passivity Approach
(91)
Pose Control Law and Stability Analysis
Error SystemEstimation
Theorem 3If , then the equilibrium point for theclosed-loop system (86) and (91) is asymptotic stable.
Pose Control & Estimation Error Systems
Error SystemPose Control
Lyapunov Function Candidate(88) (92)
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Fig. 28: Block Diagram of Pose Control with Visual Motion Observer
Pose Control Law
Pose Control with Visual Motion Observer
Pose Controller
RRBM& Camera
Model
Pose Control & Estimation Error Systems
Error SystemPose Control
Visual Motion Observer
Motion ControllerPose
Fig. 30: Block Diagram of Pose Control with Visual Motion Observer
Velocity InputVisual
Observer
Pose Computation
toto
Pose ControlError System Pose Controller
Fig. 29: Block Diagram of Pose Controller
32
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Outline
・ Relative Rigid Body Motion
・ Prologue – Cooperative Control
・ Pose Control with VMO
・ Visual Motion Observer (VMO)
・ Robot Control with VMO
・ Epilogue – Cooperative Control with VMO
33Fujita LaboratoryTokyo Institute of Technology
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ControllerPose
Pose Control with Visual Motion Observer
Manipulator Dynamics
Fig. 30: Block Diagram of Pose Control with Visual Motion Observer
MotionVisual
Observer
No Dynamics
Manipulator Dynamics
: Inertia Matrix
: Gravity Vector
: Input Torque: Coriolis Matrix : Joint Angle
(93)
:Skew-symmetric
Fig. 31: 2DOF Manipulator
1q
2q
X
Y
1r 2r
1m
2m 2I
1I
1l
2lExample : Two Degree of Freedom Planar Manipulator
Passivity of Manipulator Dynamics
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Fig. 33: Block Diagram of Manipulator
Fig. 32: Block Diagram of Pose Control with Visual Motion Observer & Manipulator
ControllerPose
MotionVisual
Observer
Manipulator Dynamics(94):Disturbance InputBody Velocity of Vision Camera
:Body Manipulator(95)
ManipulatorDynamics
Joint Velocity Error
Jacobian
Desired JointVelocity
Joint Velocity Error
Vision Camera Velocityfrom Pose Controller
(97)
(96)
Desired
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Fig. 32: Block Diagram of Pose Control with Visual Motion Observer & Manipulator
Fig. 34: Block Diagram of Robot Controller
ControllerRobot
ControllerPose
MotionVisual
Observer
Control
Passivation
Robot ControllerPassivation Control
(98):Input for Joint Velocity Error
Joint Velocity Error System
(99)
Fig. 35: Block Diagram of Joint Velocity Error System
Control
Passi-Robot Controller
Joint VelocityError System
vation
PassiveIf
Passivation Control
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Fig. 36: Block Diagram of Error System for Robot Control with Visual Motion Observer
Error System
Error System for Robot Control with Visual Motion Observer
Error System for Robot Control with Visual Motion Observer
(100)
State Input Disturbance
Error Systems
Pose Control& Estimation
Error SystemJoint Velocity
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Fig. 36: Block Diagram of Error System for Robot Control with Visual Motion Observer
Passivity of Error System for Robot Control with Visual Motion Observer
Error System
Error Systems
Pose Control& Estimation
Error SystemJoint Velocity
Passive
Lemma 5If , then the error system for robot controlwith the visual motion observer (100) satisfies
(101)
where ,and is a positive scalar.
Storage Function
(102)
Passive
38
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Differentiating (102) w.r.t. time yields
Skew-symmetricMatrix
Skew-symmetricMatrix
(103)
(Q.E.D.)
Integrating (103) from 0 to T, we obtain
where is a positive scalar that only depends on the initial states of, and .
Passivity of Error System for Robot Control with Visual Motion Observer
(Proof)
(104)
Joint Velocity Error System(Manipulator Dynamics)
Pose ControlError System
EstimationError System
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Fig. 37: Block Diagram of Robot Control with Visual Motion Observer
Robot Control Law with Visual Motion Observer
(105)
:Gain for Joint Velocity Error:Gain for Pose Control Error:Gain for Estimation Error
Robot Control Law
Robot Control Law with Visual Motion Observer : Passivity Approach
Control
Error System
Error Systems
Pose Control& Estimation
Error SystemJoint Velocity
40
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Fig. 37: Block Diagram of Robot Control with Visual Motion Observer
Stability Analysis for Robot Control with Visual Motion Observer
Theorem 4If , then the equilibrium point for the closed-loop system (100) and (105) is asymptotic stable.
Lyapunov Function Candidate
The equilibrium point is asymptotically stable.
Fig. 38: Concept ofLyapunov Function
Robot Control Law
Error System
(106)
(107)
Error Systems
Pose Control& Estimation
Error SystemJoint Velocity
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Outline
・ Relative Rigid Body Motion
・ Prologue – Cooperative Control
・ Pose Control with VMO
・ Visual Motion Observer (VMO)
・ Robot Control with VMO
・ Epilogue – Cooperative Control with VMO
42
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Rigid Body Motion in SE(3)
43
(24)
Pose (Position and Orientation)
(27)Passive
Rigid BodyMotion to
Lemma 1:
Fig. 11: Attitude Synchronization
Review: Attitude Synchronization
Attitude Synchronization(33)
Fig. 38: Coordinates of Multi-rigid Body in SE(3)
Relative Pose(107)
Control Input(35)
(36)
Fig. 10: Block Diagram of Passivity of Rigid Body Motion
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Pose Synchronization
Consider the n rigid bodies represented by (27). Then a group of rigid bodies is said to achieve pose synchronization, when all rigid bodies converge to the same pose between the rigid bodies.
Pose Synchronization
Pose Synchronization
Relative Pose
(109)
(108)
(77)
Fig. 39: Pose Synchronization
(24)
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Control Input for Pose Synchronization
Control Input for Pose Synchronization
Fig. 39: Pose Synchronization
(110)Relative Pose
Fig. 40: Block Diagram of Pose Synchronization
i-th Pose Synchronization Controller
i-th RigidBody Motion
j-th RigidBody Motion
Control Input for Pose Control under the Condition:
(91)
Sum of Input for Pose Control with Neighbors
(110)
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Pose Synchronization
Theorem 4Consider the n rigid bodies represented by (27). Then, under the assumptions A1 and A2, the velocity input (110) achieves pose synchronization in the sense of (108).
Position
Lyapunov Function Candidate(111)
Orientationare uniquely defined by strongly connected graphs.
Fig. 39: Pose Synchronization
i-th RigidBody Motion
i-th Pose Synchronization
Controller
j-th RigidBody Motion
j-th Pose Synchronization
Controller
Fig. 41: Block Diagram of Pose Synchronization
Interaction
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Motion ControllerPose
Fig. 30: Block Diagram of Pose Control with Visual Motion Observer
Visual
Observer
Control Input for Pose Synchronization
(110)
Visual Motion Observer× : not measurableRelative Pose
Fig. 39: Pose Synchronization
Fig. 41: Block Diagram of Pose Synchronization
Pose Synchronization with Visual Motion Observer
i-th RigidBody Motion
i-th Pose Synchronization
Controller
×
not measurablenot measurable
Pose Control with Visual Motion Observer
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Motion ControllerPose
Fig. 30: Block Diagram of Pose Control with Visual Motion Observer
Visual
Observer
Motioni-th Visual
Observeri-th Rigid
Body Motion
i-th Pose Synchronization
Controller
Fig. 42: Block Diagram of Pose Synchronization with Visual Motion Observer
Pose Synchronization with Visual Motion Observeri-th PoseCompu-tation
Estimated Relative PoseEstimation ErrorEstimation Error Vector
(112)
Image Information
RRBM CameraVision
Relative Rigid Body Motion
Pose Control with Visual Motion Observer
(RRBM)
(113) Fig. 18: Block Diagram of Relative RigidBody Motion with Vision Camera
i-thVision
Camera
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Fig. 42: Block Diagram of Pose Synchronization with Visual Motion Observer
49
Motioni-th Visual
Observeri-th Rigid
Body Motion
i-th Pose Synchronization
Controller
i-th PoseCompu-tation
Pose Synchronization with Visual Motion Observer
Interaction between Multiple Rigid Bodies
Motioni-th Visual
Observerand Pose Synchronization
i-th Pose Computation
Controller
Fig. 43: Block Diagram of Pose Synchronization with Visual Motion Observerin 2 Rigid Bodies
Interaction
Motionj-th Visual
Observerand Pose Synchronization
j-th Pose Computation
Controller
i-thVision
Camera
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Computationi-th Posei-th
Observer
50
Fig. 44: Block Diagram of i-th Multi-visual Motion Observer
Multi-visual Motion Observer
Pose Synchronization Control Law Pose Synchronization Control Law
(115)
VisualMotion
ObserverVisualMotion
Observer
Visual Motion Observer
Fig. 42: Block Diagram of Pose Synchronization with Visual Motion Observer
Fig. 38: Coordinates of Multi-rigid Body in SE(3)
(114)
Pose Synchronization with Visual Motion Observer
with Visual Motion Observer
PoseCompu-tation
PoseCompu-tation
i-th Pose Synchronization
Controller
Motioni-th Visual
Observeri-th Rigid
Body Motion
i-th Pose Synchronization
Controller
i-th PoseCompu-tation
i-thVision
Camera
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Outline
・ Relative Rigid Body Motion
・ Prologue – Cooperative Control
・ Pose Control with VMO
・ Visual Motion Observer (VMO)
・ Robot Control with VMO
・ Epilogue – Cooperative Control with VMO
51